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Homogenization of a mean field game system in the small noise limit

Annalisa Cesaroni, Nicolas Dirr, Claudio Marchi

TL;DR

It is shown that in general the effective system loses the MFG structure.

Abstract

This paper concerns the simultaneous effect of homogenization and of the small noise limit for a $2^{\textrm {nd}}$ order mean field games (MFG) system with local coupling and quadratic Hamiltonian. We show under some additional assumptions that the solutions of our system converge to a solution of an effective $1^{\textrm {st}}$ order system whose effective operators are defined through a cell problem which is a $2^{\textrm {nd}}$ order system of ergodic MFG type. We provide several properties of the effective operators and we show that in general the effective system looses the MFG structure.

Homogenization of a mean field game system in the small noise limit

TL;DR

It is shown that in general the effective system loses the MFG structure.

Abstract

This paper concerns the simultaneous effect of homogenization and of the small noise limit for a order mean field games (MFG) system with local coupling and quadratic Hamiltonian. We show under some additional assumptions that the solutions of our system converge to a solution of an effective order system whose effective operators are defined through a cell problem which is a order system of ergodic MFG type. We provide several properties of the effective operators and we show that in general the effective system looses the MFG structure.

Paper Structure

This paper contains 18 sections, 19 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Standing assumptions
  3. Formal asymptotic expansions
  4. Other asymptotic expansions
  5. Time-dependent asymptotic expansion and $\epsilon$-correction to initial data
  6. Case of finite noise
  7. Case of non local coupling
  8. Cell problem, effective Hamiltonian and effective vector field.
  9. Regularity properties of the effective operators
  10. Variation of $\bar{H}$ with respect to $P$
  11. Variation of $\bar{H}$ with respect to $\alpha$
  12. Regularity properties of $\bar{b}$
  13. Qualitative properties of the effective operators
  14. A case in which the limit system is not MFG
  15. Convergence for affine-constant initial data
  16. ...and 3 more sections

Key Result

Proposition 2.1

For $\epsilon=\frac{1}{k}$ ($k\in\mathbb{N}$), there exists a unique classical solution to systemintro.

Theorems & Definitions (37)

  • Proposition 2.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • proof
  • Proposition 5.1
  • proof
  • ...and 27 more